Snapshots of early experimenting with honeycombs in the upper half space model.

Snapshots of early experimenting with honeycombs in the upper half space model.  It will take some effort, but I'd eventually like to see 3D prints using this model.

http://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model





Comments

  1. Anton SherwoodSeptember 7, 2013 at 12:01 PM

    Great start!  I wish I had done it ;)

    ReplyDelete
  2. Owen MareshSeptember 7, 2013 at 3:29 PM

    I wonder what properties resistor networks or superconductors fabricated in such patterns would have.

    ReplyDelete
  3. John BaezSeptember 7, 2013 at 8:44 PM

    I like these!  Could you describe a couple of these mathematically so I could put it on the new American Mathematical Society "Visual Insight" blog?  (Picture that illustrate math ideas.)

    ReplyDelete
  4. John BaezSeptember 7, 2013 at 8:46 PM

    Oh, I downloaded one and the filename said what it was: {6,3,3}.  Would you be willing to sign a form saying I can put it on the AMS blog?  (Sorry it's so bureaucratic...)  Of course you'd be credited.

    ReplyDelete
  5. Roice NelsonSeptember 7, 2013 at 9:30 PM

    Hi John Baez.  Absolutely, I'm happy to sign the needed form.  I'm also happy to tune these as needed.  I can generate them with clear backgrounds for instance, if that would be better.  I could also create models with more resolution (smoother looking edges), or less culling.

    As for descriptions, sounds like you might be good now, but here's some quick info.

    Two of the pics are from the {5,3,6} honeycomb, which has ideal dodecahedra as cells, 6 around each edge.  All the vertices live at infinity, or the z=0 plane in the upper half space model.  The vertex figure is a {3,6} tiling, so an infinite number of cells meet at each vertex.

    As you saw, the other three are from the {6,3,3}, which have euclidean hexagonal tilings inscribed in horospheres as cells.  Since the cells have an infinite number of facets, they've been culled out towards the boundary.

    Both are flavors of the more exotic honeycombs Coxeter enumerated by allowing the cells and/or vertex figures to become infinite, but where the fundamental region of the symmetry group is still finite.

    (I've done some pics of honeycombs where this last restriction is relaxed as well, and I think they turned out quite striking.  I'll forward those along, in case you might like to use them too.)

    Hope this is helpful, and a useful level of description.

    ReplyDelete

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